The digital Fourier transform (DFT) is one of the most efficient ways to determine a signal's frequency domain characteristics. Analysis of the frequency characteristics using DFT is very useful in wide variety of fields such as analysis of the vibration in a mechanical device or measuring distortion of an electronic circuit. The fast Fourier transform (FFT) process is an important component of the DFT process. The DFT process is well known and widely used. The origin of the FFT process is traced back to Carl Friedrich Gauss, the eminent German mathematician, and the history of the FFT is described by Michael T. Heidemann et al. in "Gauss and the History of the Fast Fourier Transform", IEEE ASSP Magazine, pp. 14-21, Oct. 1984.
As is well known, an analog signal is first digitized into discrete time data of N points (numbers) where N is a positive integer. Then the FFT algorithm or other DFT algorithm processes the data to obtain corresponding digital data in frequency domain. Using N sampled data points which are sampled each t over an observation period of T (T=t N), DFT algorithm computes each Fourier spectral component at frequencies of 0 (DC), 1/T, 2/T, . . . , M/T. M is (N-2)/2 for even N and (N-1)/2 for odd N.
If the measured signal's frequency components happen to coincide with the above-mentioned discrete frequencies, the computed frequency components are correct in both their amplitudes and phases without any further processing, as is shown in FIG. 4A and FIG. 4B. In FIG. 4A, the bold line 4-1 depicts the measured analog signal and small dots 4-3 depicts sampled digital data. FIG. 4B shows the corresponding continuous Analog Fourier Transformed (AFTed) data 4-5 and Digital Fourier Transformed (DFTed) data 4-7 in arrowed vertical lines and dots respectively.
However, if the measured signal has frequency components other than 0(DC), 1/T, 2/T, . . . , M/T in a frequency range equal to or below 1/(2t), the amplitudes of corresponding AFTed data and DFTed data will not be the same as shown in FIG. 5B. The DFT process distorts the amplitude of the actual frequency components and creates additional frequency components around the true frequency components. For example, a continuous analog signal 5-1 has sample data points 5-3 as is shown in FIG. 5A. The DFT processes the sampled data points 5-3 via a rectangular window. FIG. 5B shows the discrepancy between the accurate AFT and the DFT data 5-6 shown in FIG. 5B.
A variety of windows other than rectangular windows have been used to alleviate above-mentioned errors. One of such windows utilizes Hanning window function shown in FIG. 6A and its Fourier transform as shown in FIG. 6B.
The analog signal input shown in FIG. 5A is multiplied by Hanning window function shown in FIG. 6A to produce signal 7-1 and sampled as shown in FIG. 7A. The data samples are DFTed as shown in FIG. 7B. The envelope of the AFT of the input signal 7-7 is shown in FIG. 7B as well as the DFT of the sampled data 7-5. Although the DFTed data 7-5 resides on the envelope of the AFT data 7-7, the DFT data is not available for the entire envelope. Importantly, the peaks of the AFT data are not reflected by the DFT data 7-5. This example illustrates one of the problems of the prior art. In actual practice, the input signal is multiplied by the Hanning window after the input signal is sampled. However, the above description has been given to make it easier to understand. Although many other windows are well known, further description of each is omitted here.
Level errors, leakages and windowings are treated by E. O. Brigham in Chapters 6 and 9 of "The Fast Fourier Transform", Prentice-Hall, Inc., 1974.
Windows have negative effects, however. Generally speaking, higher level accuracy accompanies lower frequency resolution and wider equivalent noise bandwidth. As examples, the Hanning window of 1.5 dB level accuracy gives 1.5.times.(1/T) in equivalent noise bandwidth and a flat top window, as shown in FIG. 8. The window can improve level accuracy to 0.1 dB but deteriorate equivalent noise bandwidth to 3.5.times.(1/T). The flat top window is described in pages 2-14 of Hewlett-Packard Journal, September 1978 and hereby incorporated into this disclosure.
Another negative effect is leakages in DFTed spectra. When a signal contains spectral components other than 0 (DC), 1/T, 2T, 3T, . . . , M/T, the DFTed data contains several spectral components that correspond to a single component in the measured data.
Those leakage spectra components group together. When shown on a CRT they are bothersome, but not a catastropic failure. However, many devices such as integrated circuit testers require precise Fourier spectrum analysis. The leakage spectra components of the prior art cannot be tolerated.